# Split interval

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In [topology](/source/topology), the '''split interval''', or '''double arrow space''', is a [topological space](/source/topological_space) that results from splitting each point in a [closed interval](/source/closed_interval) into two adjacent points and giving the resulting ordered set the [order topology](/source/order_topology).  It satisfies various interesting properties and serves as a useful counterexample in [general topology](/source/general_topology).

== Definition ==

The '''split interval''' can be defined as the [lexicographic product](/source/lexicographical_order) <math>[0, 1] \times\{0, 1\}</math> equipped with the [order topology](/source/order_topology).<ref>{{Citation |last=Todorcevic |first=Stevo |authorlink=Stevo Todorčević|date=6 July 1999 |title=Compact subsets of the first Baire class |journal=Journal of the American Mathematical Society |volume=12 |issue=4 |pages=1179–1212 |doi=10.1090/S0894-0347-99-00312-4|doi-access=free }}</ref>  Equivalently, the space can be constructed by taking the closed interval <math>[0,1]</math> with its usual order, splitting each point <math>a</math> into two adjacent points <math>a^-<a^+</math>, and giving the resulting linearly ordered set the order topology.<ref>Fremlin, section 419L</ref>  The space is also known as the '''double arrow space''',<ref>Arhangel'skii, p. 39</ref><ref>{{cite web |last1=Ma |first1=Dan |title=The Lexicographic Order and The Double Arrow Space |date=8 October 2009 |url=https://dantopology.wordpress.com/2009/10/07/the-lexicographic-order-and-the-double-arrow-space}}</ref> '''Alexandrov double arrow space''' or '''two arrows space'''.

The space above is a [linearly ordered topological space](/source/linearly_ordered_topological_space) with two isolated points, <math>(0,0)</math> and <math>(1,1)</math> in the lexicographic product.  Some authors<ref>Steen & Seebach, counterexample #95, under the name of '''weak parallel line topology'''</ref><ref>Engelking, example 3.10.C</ref> take as definition the same space without the two isolated points.  (In the point splitting description this corresponds to not splitting the endpoints <math>0</math> and <math>1</math> of the interval.)  The resulting space has essentially the same properties.

The double arrow space is a subspace of the [lexicographically ordered unit square](/source/Lexicographic_order_topology_on_the_unit_square).  If we ignore the isolated points, a [base](/source/base_(topology)) for the double arrow space topology consists of all sets of the form <math>((a,b]\times\{0\}) \cup ([a,b)\times\{1\})</math> with <math>a<b</math>.  (In the point splitting description these are the [clopen](/source/clopen) intervals of the form <math>[a^+,b^-]=(a^-,b^+)</math>, which are simultaneously closed intervals and open intervals.)  The lower subspace <math>(0,1]\times\{0\}</math> is [homeomorphic](/source/homeomorphic) to the [Sorgenfrey line](/source/Sorgenfrey_line) with half-open intervals to the left as a base for the topology, and the upper subspace <math>[0,1)\times\{1\}</math> is homeomorphic to the Sorgenfrey line with half-open intervals to the right as a base, like two parallel arrows going in opposite directions, hence the name.

== Properties ==

The split interval <math>X</math> is a [zero-dimensional](/source/zero-dimensional) [compact](/source/compact_(topology)) [Hausdorff space](/source/Hausdorff_space).  It is a [linearly ordered topological space](/source/linearly_ordered_topological_space) that is [separable](/source/separable_(topology)) but not [second countable](/source/second_countable), hence not [metrizable](/source/metrizable); its metrizable subspaces are all countable.

It is [hereditarily Lindelöf](/source/hereditarily_Lindel%C3%B6f), hereditarily separable, and [perfectly normal](/source/perfectly_normal_space) (T<sub>6</sub>).  But the product <math>X\times X</math> of the space with itself is not even [hereditarily normal](/source/hereditarily_normal_space) (T<sub>5</sub>), as it contains a copy of the [Sorgenfrey plane](/source/Sorgenfrey_plane), which is not [normal](/source/normal_space).

All [compact](/source/compactness), separable ordered spaces are order-isomorphic to a subset of the split interval.<ref>{{Citation |last=Ostaszewski |first=A. J. |date=February 1974 |title=A Characterization of Compact, Separable, Ordered Spaces |journal=Journal of the London Mathematical Society |volume=s2-7 |issue=4 |pages=758–760 |doi=10.1112/jlms/s2-7.4.758}}</ref>

== See also ==

* {{annotated link|List of topologies}}

== Notes ==
{{reflist}}

== References ==

* [Arhangel'skii, A.V.](/source/Alexander_Arhangelskii) and Sklyarenko, E.G.., ''General Topology II'', Springer-Verlag, New York (1996) {{isbn|978-3-642-77032-6}}
* [Engelking, Ryszard](/source/Ryszard_Engelking), ''General Topology'', Heldermann Verlag Berlin, 1989. {{ISBN|3-88538-006-4}}
* {{citation|first=D.H.|last=Fremlin|title=Measure Theory, Volume 4|publisher=Torres Fremlin|year=2003|isbn=0-9538129-4-4}}
* {{Cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[Counterexamples in Topology](/source/Counterexamples_in_Topology) | orig-year=1978 | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=Berlin, New York | edition=[Dover](/source/Dover_Publications) reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446  | year=1995 }}

Category:Topological spaces

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